Finite Element Simulation of Dense Wire Packings

A finite element program is presented to simulate the process of packing and coiling elastic wires in two- and three-dimensional confining cavities. The wire is represented by third order beam elements and embedded into a corotational formulation to capture the geometric nonlinearity resulting from large rotations and deformations. The hyperbolic equations of motion are integrated in time using two different integration methods from the Newmark family: an implicit iterative Newton-Raphson line search solver, and an explicit predictor-corrector scheme, both with adaptive time stepping. These two approaches reveal fundamentally different suitability for the problem of strongly self-interacting bodies found in densely packed cavities. Generalizing the spherical confinement symmetry investigated in recent studies, the packing of a wire in hard ellipsoidal cavities is simulated in the frictionless elastic limit. Evidence is given that packings in oblate spheroids and scalene ellipsoids are energetically preferred to spheres.

💡 Research Summary

This paper presents a comprehensive finite‑element framework for simulating the dense packing and coiling of elastic wires inside both two‑ and three‑dimensional confining cavities. The wire is discretized with third‑order beam elements based on Reddy’s third‑order beam theory (RBT), which incorporates a quadratic transverse shear stress distribution and thus avoids shear locking while accurately representing bending, shear, and torsional deformations. Each node carries six degrees of freedom—three translational displacements and three rotational parameters—represented internally by unit quaternions and local triads to prevent singularities associated with large Euler‑angle rotations.

Geometric nonlinearity is handled through a corotational (CR) formulation. For each element a local rotating reference frame is defined, reducing the twelve global degrees of freedom to seven local deformation variables (one axial stretch and six rotation components). The element’s constant linear stiffness matrix in the corotated frame (Ķₑ) is transformed to the global, configuration‑dependent stiffness matrix Kₑ(uₑ) via a transformation matrix Fₑ(uₑ). The internal force vector f_int,e = Fₑᵀ Ķₑ ûₑ and the consistent tangent stiffness Kₜ,e = ∂Kₑ/∂uₑ = Kₑ + K_σ,e are computed at each iteration, where K_σ,e captures the geometric stiffness arising from large deformations.

Time integration of the hyperbolic equations of motion M ü + C u̇ + f_int(u) = f_ext(u) is performed using two Newmark‑family schemes: (1) an implicit Newton‑Raphson line‑search method, and (2) an explicit predictor‑corrector scheme. Both employ adaptive time stepping. The implicit approach offers unconditional stability and high accuracy but requires solving a nonlinear system at each step, which becomes prohibitively expensive when wire‑wire contacts proliferate in dense packings. The explicit method avoids matrix assembly in every step, making it far more efficient for strongly self‑interacting configurations, albeit with a smaller stable time step. Comparative tests demonstrate that the explicit scheme dramatically reduces computational time while still capturing the essential dynamics of tightly packed wires.

Beyond reproducing earlier results for spherical cavities, the authors extend the study to hard ellipsoidal confinements, including oblate spheroids and scalene ellipsoids. The cavity surface is modeled as a rigid, frictionless barrier that exerts a repulsive elastic force upon contact. Simulations reveal that wires packed in non‑spherical cavities attain lower total elastic energy than in spherical ones. In oblate spheroids the wire preferentially aligns along the flattened plane, reducing bending curvature; in scalene ellipsoids the wire adopts asymmetric winding patterns that better accommodate the cavity’s anisotropic geometry. These findings suggest that, in the frictionless elastic limit, non‑spherical confinement is energetically favorable for dense wire packings.

The paper also discusses limitations and future work. Current simulations neglect friction and plasticity, assume perfectly rigid cavity walls, and treat only a single wire. Extending the framework to include frictional contact, material yielding, dynamic cavity deformation (e.g., vascular or aneurysm models), and multiple interacting wires would broaden its applicability to biomedical, engineering, and materials‑science problems. The authors propose incorporating advanced contact detection algorithms, adaptive mesh refinement, and experimental validation as next steps. Overall, the work delivers a robust, high‑fidelity computational tool that bridges the gap between discrete‑element models and continuum finite‑element analyses for complex, densely packed slender structures.