Discrete Differential Geometry for Simulating Nonlinear Behaviors of Flexible Systems: A Survey

TL;DR

This survey articulates how Discrete Differential Geometry (DDG) offers a geometry-first, structure-preserving alternative to conventional discretizations for nonlinear flexible systems. By discretizing geometry on meshes and employing energy-based formulations with differentiable operators such as DEC, DDG enables robust large-deformation simulations for 1D rods/ribbons and 2D plates/shells, as well as multiphysics couplings including frictional contact, magneto-elastic actuation, and fluid–solid interaction. It catalogs representative DDG models (e.g., DER) and extends to applications in mechanics, bioinspired morphogenesis, functional devices, and robotics, highlighting differentiable design, inverse problems, and real-time control. The paper also outlines opportunities in multiphysics integration, differentiable digital twins, and scalable GPU-enabled solvers to foster adoption in engineering practice and digital twin ecosystems.

Abstract

Flexible slender structures such as rods, ribbons, plates, and shells exhibit extreme nonlinear responses bending, twisting, buckling, wrinkling, and self contact, that defy conventional simulation frameworks. Discrete Differential Geometry (DDG) has emerged as a geometry first, structure preserving paradigm for modeling such behaviors. Unlike finite element or mass spring methods, DDG discretizes geometry rather than governing equations, allowing curvature, twist, and strain to be defined directly on meshes. This approach yields robust large deformation dynamics, accurate handling of contact, and differentiability essential for inverse design and learning based control. This review consolidates the rapidly expanding landscape of DDG models across 1D and 2D systems, including discrete elastic rods, ribbons, plates, and shells, as well as multiphysics extensions to contact, magnetic actuation, and fluid structure interaction. We synthesize applications spanning mechanics of nonlinear instabilities, biological morphogenesis, functional structures and devices, and robotics from manipulation to soft machines. Compared with established approaches, DDG offers a unique balance of geometric fidelity, computational efficiency, and algorithmic differentiability, bridging continuum rigor with real time, contact rich performance. We conclude by outlining opportunities for multiphysics coupling, hybrid physics data pipelines, and scalable GPU accelerated solvers, and by emphasizing DDG role in enabling digital twins, sim to real transfer, and intelligent design of next generation flexible systems.

Figures (8)

• Figure 1: Overview of DDG-based simulations and applications. (a) Representative structures: (a1) rods choi2021implicit; (a2) ribbons huang2020shear; (a3) plates bouaziz2023projective; (a4) shells huang2024discrete; (a5) gridshells huang2021numerical. (b) DDG framework: (b1) discrete geometry for slender structures in 1D and 2D; (b2) vertices and material frames enabling geometric quantities (e.g., curvature/dihedral) via DDG operators; (b3) coupling DDG with mechanics—energy assembly and force/Hessian evaluation—to obtain the equations of motion (EOMs). (c–g) Applications: mechanics studies -- (c1) coiling of rods jawed2014coiling, (c2) propulsion of helical filaments jawed2015propulsion; natural systems -- (d1) leaf shape liang2009shape, (d2) bacterial navigation huang2020numerical; flexible electronics -- (e1) wireless sensors jang2017self, (e2) nanotubes' pattern k2018patterns; robotic manipulation -- (f1) knot tying tong2024dlodeployment, (f2) paper folding choi2025folding; soft robotics -- (g1) rolling robots huang2020dynamic, (g2) jumping robots huang2019highly.
• Figure 2: DDG notations and operators for 1D rods, 2D surfaces, and multi-physics modelling including frictional contact, magneto-elastic, and fluid-solid interaction. (a) Geometry of a discrete 1D structure: the centerline is described by nodes $\mathbf x_i$; material directors $\mathbf m_1^i$, $\mathbf m_2^i$ and the tangent $\mathbf t^i$ encode bending curvature $\kappa_{1, i}$, $\kappa_{2, i}$ and twist $\tau_i$. (b) Discrete curvature operator: curvature is obtained from the turning angle between consecutive tangents $\mathbf t^{i-1}$ and $\mathbf t^i$ (parallel transport formulation); (c) Discrete schematic of a 2D structure; (d) Mid-edge operator for 2D meshes: curvature on a triangulated surface is evaluated using mid-edge quantities defined on neighboring faces. (e) Contact schematic for a pair of bodies with signed distance gap $\Delta$ and tangential relative velocity $\mathbf v_\textrm{rel}$. Frictional contact responses $\mathbf F_c$ and $\mathbf F_{fr}$ can be computed based on those. (f) Illustration of the deformation of a magnetized filament. (g) The schematic of fluid-solid interaction for a filament in the fluid fields.
• Figure 3: Applications of DDG-based simulation to mechanics study. DDG models enable quantitative analysis of nonlinear behaviors across rods, ribbons, and shells. Examples include: (a) buckling of an elastic strip and the effects of asymmetric boundary conditions on its transition wang2024transientgiudici2025transient; (b) bifurcation-driven shape evolution of a ribbon huang2020shear; (c) magneto-elastic coupling in hard-magnetic helical filaments sano2022kirchhoff; (d) pattern selection in rods deposited on a moving substrate jawed2014coiling; (g) indentation-induced buckling of a hemispherical gridshell huang2022numerical and (e) equilibrium shapes of an over-curved elastic ring korner2021simple.
• Figure 4: Applications of DDG-based simulation to natural systems. DDG models reveal biomechanical mechanisms underlying bioinspired phenomena. Shown are: (a) rotation of a bacterial flagellum in a viscous medium (image adapted from yang2024ceanimonas); (b) DDG-based simulation of flagellar dynamics jawed2017dynamics; (c) navigation and control leveraging elastic–hydrodynamic coupling huang2020numerical; (d) bundling under simultaneous rotation of multiple flagella tong2023fully; (e) ciliary locomotion gu2020magnetic; (f) leaf morphogenesis from differential growth liang2009shape; (g) gut looping driven by mesentery–tube growth mismatch savin2011growth; (h) emergent entanglement–disentanglement in worm collectives patil2023ultrafast.
• Figure 5: DDG-based simulations for functional structures and opportunities for device concepts. DDG-based simulations have shown accuracy on representative functional structures: (a) beam and rod systems, exemplified by rod packaging jung2025entanglement; (b) folding and cutting systems, exemplified by meta-ribbons huang2024integration; (c) topology-reinforced systems, exemplified by knitted structures ding2024unravelling. These structures motivate devices that were not analyzed with DDG in the cited works but are strong candidates for DDG-based modeling: flexible electronics, including (d1) stretchable electrical interconnects jang2017self and (d2) serpentine microstructure stretchable electronics zhang2013buckling; foldable systems, including (e1) kirigami-inspired foldable wings faber2018bioinspired and (e2) reconfigurable opto-NEMS kirigami chen2021electromechanically; artificial muscles, including (f1) Lyocell-based textile actuators maziz2017knitting and (f2) knotted liquid crystal elastomers(LCE) actuators chen2024knotted.