Equilibrium configurations of thin elastic rods have been of interest since the times of Euler. The mathematical modeling of these deformable structures has been reduced from three dimensions to a one-dimensional problem for the centerline of the rod \(u:I\to \mathbb{R}^{3}\); see [8, 22, 29, 32, 37]. In the bending regime the rod is inextensible so that \(|u'|=1\) on I. Considering a circular cross section and omitting twist contributions, the elastic energy reduces to the functional
$$ E_{\mathrm{bend}}[u]=\frac{\kappa}{2} \int _{I} \bigl\vert u''(x) \bigr\vert ^{2}{\,{ \mathrm {d}}x} $$
for a parameter \(\kappa >0\) that describes the bending rigidity. Elasticae, i.e., rods of minimal bending energy, can be stated explicitly, e.g., for periodic boundary conditions [33, 34]. Applications of elastic thin rods include DNA modeling [1, 27, 43], the movement of actin filaments in cells [36] or of thin microswimmers [41], the fabrication of textiles [30], and investigating the reach of a rod injected into a cylinder [38].
To obtain minimally bent elastic rods, the bending energy can be reduced by a gradient-flow approach. This method can be used for analytic considerations, cf. for instance [19, 26, 34, 39, 42] and numerical computations [2–7, 13–15, 20, 21, 25, 45]. An efficient finite-element approach with an accurate treatment of the inextensibility condition can be used to find equilibria of free elastic rods [7] and self-avoiding rods [12, 13]. It can also be generalized to include twist contributions defined via torsion quantities [11]. We follow common conventions and refer to rods as elastic curves when twist contributions are omitted.
In this paper, we prpose a generalization of the existing scheme [7] to calculate elasticae of confined elastic curves. Confinements of elastic structures arise on a variety of length scales, such as DNA plasmids or biopolymers inside a cell or chamber [18, 40]. The boundary of closely packed elastic sheets or a wire in a container can be modeled as confined elastic rods in two dimensions [16, 23]. Planar settings have been addressed numerically and analytically in [24, 46], and a numerical scheme for thick elastic curves in containers is devised in [44].
We propose an approach that can be used for rods embedded in arbitrary dimensions confined to convex domains. For the mathematical modelling, we use a gradient flow to minimize the bending energy. The admissible rod configurations during the flow are restricted to a domain \(D\subset \mathbb{R}^{3}\).
The task to unbend a rod inside D can be translated to minimizing \(E_{\mathrm{bend}}\) among all
$$ u\in \mathcal{A}_{D} =\bigl\{ v\in H^{2}\bigl(I; \mathbb{R}^{3}\bigr)\,:v \in D \text{ and } \bigl\vert v' \bigr\vert =1\text{ a.e.},\, L_{ \mathrm{bc}}(v)=\ell _{\mathrm{bc}} \bigr\} . $$
Since the constraint \(v(x)\in D\) will be treated via a penalty approach, we also make use of the set of unconstrained curves
$$ \mathcal{A} = \bigl\{ v\in H^{2}\bigl(I;\mathbb{R}^{3} \bigr)\,: \bigl\vert v' \bigr\vert =1 \text{ a.e.},\, L_{\mathrm{bc}}(v)=\ell _{\mathrm{bc}} \bigr\} . $$
The bounded linear operator \(L_{\mathrm{bc}}:H^{2}(I;\mathbb{R}^{3}) \to \mathbb{R}^{\ell}\) realizes appropriate boundary conditions, e.g., periodic boundary conditions are imposed via \(L_{\mathrm{bc}}(v) = 0\) with
$$ L_{\mathrm{bc}}(v) = \bigl(v(b)-v(a),v'(b)-v'(a) \bigr) $$
if \(I=(a,b)\). Since we aim at the construction of an efficient numerical scheme, we restrict our considerations to those subsets D that can be written as finite intersections of simple quadratic confinements \(D_{r}\), \(r=1,2,\dots ,r_{D} \), i.e.,
$$ D= \bigcap_{r=1}^{r_{D} } D_{r}, \quad D_{r} = \bigl\{ y\in \mathbb{R}^{3}: \vert y \vert _{D_{r}}^{2} = y \cdot G_{D_{r}}y\leq 1\bigr\} $$
for symmetric positive semidefinite matrices \(G_{D_{r}} \in \mathbb{R}^{3\times 3}\). We call the finite intersection a composite quadratic confinement. Our scheme therefore excludes the sets defined by, e.g., other vector norms, such as \(D=\{y\in \mathbb{R}^{3}: |y|_{p}^{2}\leq 1\}\) with \(p\neq 2\). For ease of presentation, we often consider one set \(D_{r}\) and then omit the index r. Some basic simple quadratic confinements are the ball with radius R and \(G_{D}=I_{3}/R^{2}\), the ellipsoid with radii \(R_{1}\), \(R_{2}\), \(R_{3}\) and \((G_{D})_{ij}=\delta _{ij}/R_{i}^{2}\), or the space between two parallel planes with distance 2R with normal vector n and \(G_{D}=nn^{t}/R^{2}\). Infinite cylinders have \((G_{D})_{ii}=0\) for exactly one index i. Boxes and finite cylinders can be constructed as composite quadratic confinements. In general, any simple or composite quadratic confinement is a convex, closed, and connected set. We remark that our convergence analysis also applies to nonquadratic confinements, but the efficiency of the devised iterative scheme substantially depends on this feature.
We enforce the confinement via a potential approach, so a nonnegative term is added to the bending energy whenever the curve violates the confining restrictions. We define the potential \(V_{D}:\mathbb{R}^{3}\to \mathbb{R}\) for a simple quadratic confinement D that vanishes in D and is strictly positive on \(\mathbb{R}^{3}\backslash D\) via
$$ V_{D}(y)=\frac{1}{2} \bigl( \vert y \vert _{D}-1 \bigr)_{+}^{2}= \frac{1}{2} \vert y \vert _{D}^{2} + \frac{1}{2} V_{D}^{\mathrm{cv}}(y), $$
where the concave part \(V_{D}^{\mathrm{cv}}\) is given by the continuous function
$$ V_{D}^{\mathrm{cv}}(y)= \textstyle\begin{cases} - \vert y \vert _{D}^{2}&\text{if $y\in D$,} \\ -2 \vert y \vert _{D}+1&\text{otherwise.} \end{cases} $$
The potential is used to define the penalizing confinement energy functional
$$ E_{D}[u]= \int _{I} \bigl\vert u(x) \bigr\vert _{D}^{2}+V_{D}^{\mathrm{cv}}\bigl(u(x) \bigr){\,{ \mathrm {d}}x}, $$
which is nonnegative by the definition of the potential and zero if and only if the curve entirely lies within D. For a composite confinement defined via a family \((D_{r})_{r=1,\dots ,r_{D} }\) of simple quadratic confinements, we sum the corresponding confinement energies up, i.e.,
$$ E_{D}[u]=\sum_{r=1}^{r_{D}} E_{D_{r}}[u], \qquad V_{D}(y)=\sum _{r=1}^{r_{D} } V_{D_{r}}(y). $$
(1)
We remark that translated domains and half-spaces, e.g., \(D = \{y\in \mathbb{R}^{3}: |y-y_{D}|_{D}^{2} \le 1\}\) and \(D = \{y\in \mathbb{R}^{3}: a_{D}\cdot y \le 1 \}\), can be similarly treated.
Given \(\varepsilon >0\), a curve \(u_{\varepsilon}\in \mathcal{A}\) is called an (approximately) confined elastica if it is stationary for the functional
$$ E_{\varepsilon}[u]= E_{\mathrm{bend}}[u] +\frac{1}{2\varepsilon}E_{D}[u] $$
in the set \(\mathcal{A}\). If \(V_{D}(u_{\varepsilon})=0\) almost everywhere on I, then the rod is called an exactly confined elastica. The parameter ε determines the steepness of the quadratic well potential and defines a length-scale for the penetration depth of the curve into the space outside of D.
Considering a simple quadratic confinement \(D\subset \mathbb{R}^{3}\), we let \(V_{D}\in C^{1}(\mathbb{R}^{3};\mathbb{R})\) be the corresponding quadratic-well potential and choose \(\varepsilon >0\). Trajectories \(u\in H^{1}([0,T];L^{2}(I;\mathbb{R}^{3}))\cap L^{\infty}([0,T]; \mathcal{A})\) are defined by gradient flow evolutions. In particular, for an inner product \((\cdot ,\cdot )_{\star}\) on \(L^{2}(I;\mathbb{R}^{3})\) and an initial configuration \(u(0,x)=u_{0}(x)\), we define the temporal evolution as the solution of the time-dependent nonlinear system of partial differential equations
$$ \begin{aligned} &(\partial _{t} u,v)_{\star}+ \kappa \bigl(u'',v'' \bigr) + \varepsilon ^{-1}(u,G_{D}v) \\ &\quad = - (2\varepsilon )^{-1}\bigl(\nabla V_{D}^{\mathrm{cv}}(u),v \bigr) -\bigl( \lambda u',v'\bigr) \end{aligned} $$
(2)
for test functions \(v\in \mathcal{V}\) with a suitable set \(\mathcal{V}\) and all \(t\in [0,T]\). The function \(\lambda \in L^{1}([0,T]\times I)\) is a Lagrange multiplier associated with the arc-length condition. Confined elasticae are stationary points for (2).
For time discretization, we use backward differential quotients. Let \(\tau >0\) be the fixed time-step, and let \(k\geq 0\) be a nonnegative integer. We set \(u^{0}:=u_{0}\) and define the time step
$$ d_{t} u^{k+1}=\frac{u^{k+1}-u^{k}}{\tau}. $$
The gradient flow system is evaluated implicitly except for the concave confinement energy, which is handled explicitly due to its nonlinearity and antimonotonicity, and the Lagrange multiplier term, which is treated semiimplicitly. We hence have
$$ \begin{aligned} &\bigl(d_{t}u^{k+1},v \bigr)_{\star}+\kappa \bigl(\bigl[u^{k+1}\bigr]'',v'' \bigr) + \varepsilon ^{-1}\bigl(u^{k+1},G_{D}v \bigr) \\ &\quad = - (2\varepsilon )^{-1}\bigl(\nabla V_{D}^{\mathrm{cv}} \bigl(u^{k}\bigr),v\bigr)- \bigl( \lambda ^{k+1} \bigl[u^{k}\bigr]',v'\bigr) \end{aligned} $$
(3)
for suitable test curves \(v\in \mathcal{V}\). To ensure that the parameterization by arc-length is approximately preserved throughout the gradient flow, the constraint \(|[u^{k}]'|^{2} =1\) is linearized. This yields the first-order orthogonality condition
$$ \bigl[u^{k}\bigr]'\cdot \bigl[d_{t}u^{k+1} \bigr]'=0\quad \text{on $I$}. $$
(4)
By imposing the same condition on test curves, i.e.,
$$ \bigl[u^{k}\bigr]'\cdot v'=0\quad \text{on $I$}, $$
(5)
the Lagrange multiplier term disappears in (4). Given \(u^{0},u^{1},\ldots ,u^{k}\in H^{2}(I;\mathbb{R}^{3})\), there are unique functions \(d_{t}u^{k+1}\in H^{2}(I;\mathbb{R}^{3})\) that solve the gradient flow equation (3) with all v satisfying (5) and \(L_{\mathrm{bc}}[v]=0\). This is a direct consequence of the Lax–Milgram lemma, provided that \(v\mapsto \|v\|_{\star }+ \|v''\|\) defines an equivalent norm on the kernel of \(L_{\mathrm{bc}}\) as a closed subspace of \(H^{2}(I;\mathbb{R}^{3})\).
For numerical computations, we subdivide I into a partition \(\mathcal{P}_{h}\) of maximal length h, which can be represented by the nodes \(x_{0}< x_{1}<\cdots <x_{N}\). We use the space of piecewise cubic, globally continuously differentiable splines on \(\mathcal{P}_{h}\) as a conforming subspace \(V_{h}\subset H^{2}(I)\). On an interval \([x_{i},x_{i+1}]\), these functions are entirely defined by the values and the derivatives at the endpoints. We also employ the space of piecewise linear, globally continuous finite element functions determined by the nodal values and denote the set by \(W_{h}\). The corresponding interpolation operators are denoted as \(\mathcal{I}_{3,h}\) and \(\mathcal{I}_{1,h}\), respectively. We impose the orthogonality of \(d_{t}u_{h}^{k+1}\) and \(u_{h}^{k}\) only at the nodes. The confinement quantities are evaluated by mass lumping, so only the values
$$ (v,w)_{h}:= \int _{I}\mathcal{I}_{1,h}(v\cdot w){\,{\mathrm {d}}x} $$
at the nodes are required. In the nodal points the concavity of \(V_{D}^{\mathrm{cv}}\) is used to prove an energy monotonicity property.
We consider a controlled violation of the arclength constraint at the nodes of the partitioning determined by a parameter \(\delta _{h}\ge 0\) and define the discrete admissible set via
$$ \mathcal{A}_{h} := \bigl\{ u_{h} \in V_{h}^{3}: \bigl\vert \bigl\vert u_{h}'(x_{i}) \bigr\vert ^{2}-1 \bigr\vert \le \delta _{h},\, i=0,1,\dots ,N, \, L_{\mathrm{bc}}[u_{h}] = \ell _{\mathrm{bc}} \bigr\} . $$
The set of test functions relative to \(u_{h}\) is
$$ \mathcal{F}_{h}[u_{h}] := \bigl\{ v_{h} \in V_{h}^{3}: u_{h}'(x_{i}) \cdot v_{h}'(x_{i}) = 0, \, i=0,1,\dots ,N, \, L_{\mathrm{bc}}[v_{h}] = 0 \bigr\} . $$
We thus obtain the following fully practical numerical scheme to compute confined elasticae: Given \(u_{h}^{0} \in \mathcal{A}_{h}\), define \(u_{h}^{1},\ldots ,u_{h}^{k}\in V_{h}^{3}\) by calculating \(d_{t}u_{h}^{k+1}\in \mathcal{F}_{h}[u_{h}^{k}]\) such that
$$ \begin{aligned} &\bigl(d_{t}u_{h}^{k+1},v_{h} \bigr)_{\star}+\kappa \bigl(\bigl[u_{h}^{k} \bigr]''+ \tau \bigl[d_{t}u_{h}^{k+1} \bigr]'',v_{h}'' \bigr) +\varepsilon ^{-1}\bigl(u_{h}^{k}+ \tau d_{t}u_{h}^{k+1},G_{D}v_{h} \bigr)_{h} \\ &\quad = - (2\varepsilon )^{-1}\bigl(\nabla V_{D}^{\mathrm{cv}} \bigl(u_{h}^{k}\bigr),v_{h} \bigr)_{h} \end{aligned} $$
(6)
for all \(v_{h}\in \mathcal{F}_{h}[u_{h}^{k}]\).
The key feature of our numerical method is that it detects stationary configurations of low energy. It can in general not be guaranteed that these are global minimizers, but the stable symmetries observed in the experiments for different starting values indicate that this is often the case. Our convergence theory assumes almost global discrete minimizers. Different approaches based on working with the Euler–Lagrange equations or using additional properties of the energy functional can lead to more general convergence theories but are beyond the scope of our paper.
The remainder of the paper is structured into a first part proving the convergence of the proposed numerical scheme and into a second part that presenting the results of numerical experiments and describes confined elasticae for closed rods in balls. The numerical simulations were done in the web application Knotevolve [9], which is accessible at aam.uni-freiburg.de/knotevolve.