Iterative Force-Biased Fiber Packing

• Iterative force-biased fiber packing is an incremental strategy that sequentially updates fiber configurations under elastic, frictional, and contact forces to achieve equilibrium.
• The methodology employs advanced discretizations like finite element analysis and predictor-corrector schemes to resolve complex interactions and control force bias.
• Emerging applications range from biomedical coil design to synthetic fiber systems, highlighting the impact of friction, torsion, and geometry-induced frustration.

Iterative force-biased fiber packing denotes an incremental packing strategy in which a filament, wire, ring, or ball-chain representation is advanced stepwise and, at each step, the configuration is updated under elastic, contact, frictional, and recovery forces until a new equilibrium or minimized-energy state is reached. In the literature, this appears as quasi-static wire insertion into rigid or flexible cavities, gradual confinement by a shrinking boundary, finite element simulation of dense self-contacting beams, and post-generation overlap removal followed by explicit contact enhancement in the Altendorf-Jeulin model. Across these variants, the central controls are bending, torsion, friction, shell elasticity, contact resolution, and geometry-induced frustration [(Stoop et al., 2011); (Vetter et al., 2015); (Vetter et al., 2011); (Alben, 2021); (Grason, 2014); (Keilmann et al., 12 Sep 2025)].

1. Mechanical description of fibers and confinements

A common starting point is elastic rod theory. In spherical cavity packing, the wire is modeled as an elastic rod whose state is described by its centerline $\gamma$ and an orthonormal director frame $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$ at each cross-section, with elastic energy

$$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$

Here $A = \pi a^2$, $I = \pi a^4/4$, and $G = Y / 2(1+\nu)$. In that formulation, wires have negligible stretching due to high $Y$, so bending and torsion dominate (Stoop et al., 2011).

For flexible confinements, the wire is modeled as an extensible, untwisted Kirchhoff rod and the shell as a linearly elastic Kirchhoff–Love shell. The wire energy is

$$U_\mathrm{w} = \frac{1}{2} \int_0^L E_\mathrm{w}I\kappa^2 + GJk_3^2 + E_\mathrm{w}A\epsilon^2\,ds,$$

while the shell energy is

$$U_\mathrm{s} = \frac{1}{2} \int_{\Omega} H^{ijkl}\left( M \alpha_{ij}\alpha_{kl} + B \beta_{ij}\beta_{kl} \right)\,d\Omega,$$

with $M = E_\mathrm{s}t/(1-\nu_\mathrm{s}^2)$ and $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$0. The governing dimensionless parameters are $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$1, $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$2, $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$3, and $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$4 (Vetter et al., 2015).

Alternative discretizations are also used. Dense wire packing in arbitrary cavities employs third order beam elements within a corotational formulation, with 6 degrees of freedom per node and cubic Hermite interpolation of the centerline. Elastic rings with friction are represented by a spring-lattice model, where the bending modulus satisfies

$\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$5

so that the lattice reproduces the inextensible Euler-Bernoulli beam theory in the continuum limit [(Vetter et al., 2011); (Alben, 2021)].

2. Incremental and iterative update schemes

In these methods, packing is not treated as a single global optimization. It is treated as a sequence of force-biased equilibria. The update may be driven by injection, by gradual reduction of the confining boundary, or by iterative repositioning of already generated fibers.

System	Incremental driver	State update
Rigid spherical cavity	Wire is inserted iteratively or quasi-statically	Configuration is allowed to relax under imposed forces/moments to minimize energy, subject to contact and frictional constraints
Flexible shell	More wire is pushed in at each simulation step	New equilibrium is found by minimizing total energy $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$6, including contact/frictional forces
Altendorf-Jeulin packing	Random packing followed by iterative force application	Ball-chains are repositioned by repulsion, recovery, and contact forces

In spherical packing, the wire is discretized into $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$7 mass-points $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$8 connected by straight edges, and each edge has an associated local director frame or quaternion. Forces and moments are calculated as the negative gradient of the discretized elastic energy functional. Numerical integration is via a 6th order predictor-corrector scheme with small viscous damping for stability, which is crucial for a quasi-static or force-relaxation process (Stoop et al., 2011).

In flexible-shell packing, the simulation proceeds by finite element minimization of the total energy with Newtonian time integration and viscous damping. Each step advances the wire tip, finds a new equilibrium, applies the Coulomb friction law to contacting bodies, and updates observables such as energy, curvature, contact number, and morphology markers (Vetter et al., 2015).

Dense wire packings in two- and three-dimensional cavities introduce an explicit comparison between two Newmark-family integrators: an implicit iterative Newton-Raphson line search solver and an explicit predictor-corrector scheme, both with adaptive time stepping. For strongly self-interacting bodies in densely packed cavities, the explicit scheme outperforms implicit methods by an order of magnitude in total simulation time (Vetter et al., 2011).

Elastic rings with friction use a quasistatic, force-biased iterative procedure in which the circular boundary radius $\{\vec{d}_1,\vec{d}_2,\vec{d}_3\}$9 is reduced incrementally in small steps, each step typically multiplying $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$0 by $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$1, and equilibrium is determined by energy minimization with Newton’s method starting from the previous configuration (Alben, 2021).

The Altendorf-Jeulin model begins from fibers represented as chains of overlapping spheres generated as random walks with prescribed statistics for orientation and curvature. The model then iteratively applies repulsion forces to resolve overlaps and recovery forces to preserve internal fiber structure and maintain prescribed statistical features such as curvature and direction, yielding a hardcore fiber system (Keilmann et al., 12 Sep 2025).

3. Force bias: torsion, friction, and explicit contact control

In rigid spherical cavities, force bias can be imposed through torsional boundary conditions. In the low torsion setup, the wire can freely rotate axially at the injection point, minimizing torsional buildup and producing a bending-dominated process. In the high torsion setup, axial rotation at injection is suppressed; precurved wires are used to hinder torsion release; and counteracting axial moments are applied to prohibit twist release at the entrance. For the high torsion case, during each iteration, the net axial torque at the entrance is canceled, so torsion cannot relax and thus accumulates. Contact interactions with walls and self-contact use a repulsive potential, while friction is handled via a stick-slip Coulomb model with $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$2 and $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$3 (Stoop et al., 2011).

In flexible shells, friction itself acts as the force bias. Wire injection is modeled as an incremental, force-biased process: at each simulation step, more wire is pushed in, and the state is updated through energy minimization and contact-frictional resolution. Low friction allows easy sliding and coiling, whereas high friction impedes sliding and makes forces propagate into the shell and wire, leading to buckling and crumpling. In rigid shells, even with high friction, wire slides with minimal effect from friction; in flexible shells, friction and shell compliance give rise to a continuous phase transition between ordered and disordered morphologies. The order parameter is

$$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$4

where $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$5 measures average handedness of coiling about the principal axis $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$6 (Vetter et al., 2015).

In ring confinement, friction determines whether contacts slide or lock. The tangential component of the force at each contact never exceeds $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$7 times the normal component, and at equilibrium the static Coulomb friction law is respected. The simulation thereby produces equilibrium paths that reflect the sequentially “locked” morphology imposed by friction (Alben, 2021).

The most explicit force-biased extension appears in the contact-enhanced Altendorf-Jeulin model. There, two balls from different fibers are in contact if $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$8, and they are $$E_{el} = \frac{1}{2} \int_0^L ds \left\{ YA \left( r_3' - r_3^{0\prime} \right)^2 + YI \left[ (k_1 - k_1^0)^2 + (k_2 - k_2^0)^2 \right] + G (k_3 - k_3^0)^2 \right\}.$$9-close if $A = \pi a^2$0. A shortlist of candidate pairs is constructed so that each ball belongs to at most one contact pair per fiber pair. The added contact force is

$A = \pi a^2$1

with

$A = \pi a^2$2

and, for soft contact, a smoothed version

$A = \pi a^2$3

The total force becomes

$A = \pi a^2$4

This changes force-biased packing from overlap removal alone to overlap removal plus explicit increase of inter-fiber contacts (Keilmann et al., 12 Sep 2025).

4. Emergent morphologies and phase behavior

The rigid spherical cavity problem exhibits two principal morphologies. Low torsion packings are ordered, coil-like arrangements that layer ring structures from outside inward. High torsion packings are disordered, homogeneous, “crumpled” packings filled with many reorientations and figure-eight loops. The highest packing densities are achieved in low torsion packings for large systems, but in high torsion packings for small systems. The maximum packing fraction is

$A = \pi a^2$5

and, in the high torsion or disordered regime, the scaling behavior is

$A = \pi a^2$6

Ordered packings have a bending-energy distribution sharply peaked at low values, whereas disordered packings have a log-normal bending-energy distribution (Stoop et al., 2011).

Flexible shells display an analogous but friction-controlled distinction. At low friction, the wire densely coils into an ordered toroidal bundle with semi-ellipsoidal cross-section. At high friction, it packs into a highly disordered, hierarchic structure with frequent out-of-plane buckling and reorientation of loop axes. In the disordered regime, the bending energy, total curvature, and number of contacts obey power laws in the packed length $A = \pi a^2$7: $A = \pi a^2$8

$A = \pi a^2$9

$I = \pi a^4/4$0

Distributions of curvature and loop length are lognormal, characteristic of hierarchical structure (Vetter et al., 2015).

Elastic rings within a shrinking circular boundary display a related sequence of transitions. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. At very low friction, spiraling dominates. At moderate to high friction, bifurcations produce multiple lobed or S-shaped configurations. For very large friction coefficients, sliding is heavily suppressed and the wall-friction case becomes dominated by buckling near the wall. For intermediate friction values, the sequence often alternates between spiraling and bifurcation (Alben, 2021).

A plausible implication is that iterative force-biased packing is intrinsically path-dependent: the observed structure is selected not only by geometry and material constants, but also by whether the update sequence permits sliding, twist release, or reorientation before contacts lock.

5. Energetics, metric geometry, and packing frustration

Energetic analysis is central to these models. In the absence of torsion and for straight rods, the bending energy is

$I = \pi a^4/4$1

For spherical cavity packing, comparison is made to the DNA-packing model of Purohit et al.,

$I = \pi a^4/4$2

where

$I = \pi a^4/4$3

The dimensionless insertion force is written as $I = \pi a^4/4$4, and experimentally this is larger than analytical predictions because friction is absent from analytic bending-only models (Stoop et al., 2011).

In flexible shells, the low-friction ordered toroidal state admits an analytical approximation. The packed length is

$I = \pi a^4/4$5

with $I = \pi a^4/4$6 and $I = \pi a^4/4$7, and the bending energy is

$I = \pi a^4/4$8

These formulas formalize the ordered coiling branch against which disordered crumpling is compared (Vetter et al., 2015).

Grason’s review places such packings within a metric theory of filament organization. The distance of closest approach between neighboring filaments is not generally the Euclidean in-plane distance when filaments are not parallel; instead,

$I = \pi a^4/4$9

For slowly varying orientation texture $G = Y / 2(1+\nu)$0, the effective Gaussian curvature is

$G = Y / 2(1+\nu)$1

For a double-twist texture,

$G = Y / 2(1+\nu)$2

Positive curvature means local hexagonal packing is impossible everywhere, so geometrical frustration is required. The compatibility equation

$G = Y / 2(1+\nu)$3

links stress, disclinations, dislocations, and curvature, and the ideal topological charge of a twisted bundle is

$G = Y / 2(1+\nu)$4

The review further connects packings of twisted filament bundles to packings on positively curved surfaces, to the Thomson problem, and to fibrations of curved three-dimensional space including the Hopf fibration. After stereographic projection to $G = Y / 2(1+\nu)$5, ideal packings are only locally optimal and frustration becomes unavoidable for thick toroidal bundles (Grason, 2014).

This suggests that force-biased packing algorithms are not merely numerical devices for collision resolution; they are also mechanisms for navigating non-Euclidean incompatibility, defect entry, and the competition between locally preferred spacing and globally constrained geometry.

6. Validation, computational performance, and application domains

Validation criteria differ across implementations, but they consistently target both morphology and statistics. In the contact-enhanced Altendorf-Jeulin model, after random-walk initialization, standard packing, and contact-increased packing, the anisotropy parameter $G = Y / 2(1+\nu)$6 and curvature parameters $G = Y / 2(1+\nu)$7 are estimated and compared with desired input values. The directional parameter $G = Y / 2(1+\nu)$8 is preserved with very small deviation, reported as less than $G = Y / 2(1+\nu)$9 relative deviation for $Y$0 in all packing phases. For curvature, the deviation is less than $Y$1 for $Y$2, while $Y$3 can reach up to $Y$4, and the additional contact force does not degrade accuracy compared to the base Altendorf-Jeulin model (Keilmann et al., 12 Sep 2025).

The same study evaluates whether explicit force bias can increase contact density. The contact force substantially increases the number of inter-fiber contacts, even beyond the intersection intensities predicted by Toll’s formula in many parameter regimes. For $Y$5, $Y$6, $Y$7, $Y$8, the mean number of contacts per fiber is approximately $Y$9 in Toll’s model, approximately $$U_\mathrm{w} = \frac{1}{2} \int_0^L E_\mathrm{w}I\kappa^2 + GJk_3^2 + E_\mathrm{w}A\epsilon^2\,ds,$$0 in Altendorf-Jeulin, and approximately $$U_\mathrm{w} = \frac{1}{2} \int_0^L E_\mathrm{w}I\kappa^2 + GJk_3^2 + E_\mathrm{w}A\epsilon^2\,ds,$$1 in the contact model. The contact distribution is somewhat more narrow than the Poisson distribution predicted by Toll, and the runtime is about 10 minutes per realization for a cubic box of length 384 units (Keilmann et al., 12 Sep 2025).

In dense wire packings, computational performance is dominated by self-contact handling. The implicit Newton-Raphson solver becomes expensive and sometimes prone to non-convergence as the number of self-contact pairs grows and the potential energy landscape becomes rugged. The explicit predictor-corrector scheme avoids assembly and solution of large nonlinear systems and remains robust in dense, contact-dominated states (Vetter et al., 2011).

Application contexts recur throughout the literature. Flexible-shell packing identifies biomedical applications such as aneurysm coiling and argues that friction and shell flexibility are essential factors affecting stability and density of packed wires. The contact-enhanced Altendorf-Jeulin model is motivated by digital twins for material design or biomedicine, particularly where inter-fiber contacts influence thermal and mechanical behavior. Dense wire packing studies connect the same methodological issues to biomaterials, embolic coils, and synthetic fiber bundles [(Vetter et al., 2015); (Keilmann et al., 12 Sep 2025); (Vetter et al., 2011)].

A plausible implication is that future force-biased fiber-packing workflows will continue to couple explicit contact control, incremental insertion or confinement updates, and geometry-aware defect handling. The cited work already shows that realistic packing pathways require simultaneous treatment of elasticity, self-contact, friction, confinement compliance, and the statistical targets of the resulting fiber system.